This thread is useless without measurements
You don't need to use an SMPS (and definitely
shouldn't) to do that testing. A signal generator, capacitor, resistor and oscilloscope do a fine job. A pulse tester can also be built, usually using a signal generator (or in a pinch, a 555 timer), MOSFET, diode, capacitors, shunt resistor, and power supply. Example on my old website,
https://www.seventransistorlabs.com/tmoranwms/Elec_Magnetics.html which, the magnetic theory part I think is still fine, if a bit out of style for me these days, and I do have better and more diverse inductor and transformer design methods today (but, also which aren't worth going into detail on such a page).
After fractures (which can be glued back together with superglue or epoxy, with care), probably burrs and varnish are the next most common fault. Use a knife to scrape off the excess, and use a very flat sharpening stone or lap to grind the core faces flat. If you find only the outer limbs are grinding, congratulations -- you have an air-gapped core, and the inductivity (A_L) will be low; this is suitable for flyback converters and inductors, and unsuitable for forward converters and CMCs.
Occasionally you'll see an E-core (or other shapes) that is definitely not air-gapped, and yet doesn't have a high A_L; I haven't seen these very often (and, I seem to have lost the one example that I had found..!), but, these shapes are occasionally made in powdered-iron and related materials, with lower permeability, so can be used ungapped just as any powder toroid.
As for leakage, or coupling coefficient, note this also depends upon the relative positioning of windings. The core is not an ideal magnetic path (and even if it were, it wouldn't actually matter--), and there are plenty of flux paths outside of it, literally leaking around; wires hanging around anywhere in the space around the core, can pick up varying amounts of those paths. You maximize coupling when the windings are, as much as possible, in the same place: adjacent layers, twisted pair or bifilar, alternating layers or strands in parallel, etc.
Tim